Heat kernels, Bergman kernels, and cusp forms

TL;DR

This paper introduces a geometric approach using heat and Bergman kernel analysis to derive bounds for cusp forms across various weights, advancing the understanding of their analytic properties.

Contribution

It presents a novel method applying micro-local analysis of heat and Bergman kernels to study cusp forms, bridging analytic geometry and automorphic form theory.

Findings

Derived sup-norm bounds for cusp forms of various weights

Applied micro-local analysis techniques from analytic geometry

Extended bounds to forms associated with Fuchsian groups

Abstract

In this article, we describe a geometric method to study cusp forms, which relies on heat kernel and Bergman kernel analysis. This new approach of applying techniques coming from analytic geometry is based on the micro-local analysis of the heat kernel and the Bergman kernel in \cite{bouche} and \cite{berman}, respectively, using which we derive sup-norm bounds for cusp forms of integral weight, half-integral weight, and real weight associated to a Fuchsian subgroup of first kind.